x ( t ) = x ( t + T ) = a 0 + ∑ n = 1 ∞ a n cos ( n ω 0 t ) + b n sin ( n ω 0 t ) a 0 = 1 T ∫ 0 T f ( t ) d t a n = 2 T ∫ 0 T f ( t ) cos ( n ω 0 t ) d t b n = 2 T ∫ 0 T f ( t ) sin ( n ω 0 t ) d t {\displaystyle {\begin{aligned}x(t)&=x(t+T)=a_{0}+\sum _{n=1}^{\infty }a_{n}\cos(n\omega _{0}t)+b_{n}\sin(n\omega _{0}t)\\a_{0}&={\frac {1}{T}}\int _{0}^{T}f(t)dt\\a_{n}&={\frac {2}{T}}\int _{0}^{T}f(t)\cos(n\omega _{0}t)dt\\b_{n}&={\frac {2}{T}}\int _{0}^{T}f(t)\sin(n\omega _{0}t)dt\\\end{aligned}}}
a 0 = 1 T ∫ 0 1 2 T H d t + 1 T ∫ 1 2 T T − H d t = 1 T [ H t ] | t = 0 1 2 T − 1 T [ H t ] | t = 1 2 T T = 1 T H 1 2 T − 0 − [ 1 T H T − 1 T H 1 2 T ] = H 2 − [ H − 1 2 H ] = H 2 − H 2 = 0 {\displaystyle {\begin{aligned}a_{0}&={\frac {1}{T}}\int _{0}^{{\frac {1}{2}}T}Hdt+{\frac {1}{T}}\int _{{\frac {1}{2}}T}^{T}-Hdt\\&={\frac {1}{T}}\left[Ht\right]{\bigg |}_{t=0}^{{\frac {1}{2}}T}-{\frac {1}{T}}\left[Ht\right]{\bigg |}_{t={{\frac {1}{2}}T}}^{T}\\&={\frac {1}{T}}H{\frac {1}{2}}T-0-\left[{\frac {1}{T}}HT-{\frac {1}{T}}H{\frac {1}{2}}T\right]\\&={\frac {H}{2}}-\left[H-{\frac {1}{2}}H\right]\\&={\frac {H}{2}}-{\frac {H}{2}}\\&=0\end{aligned}}}
a n = 2 T ∫ 0 1 2 T H cos ( n ω 0 t ) d t + 2 T ∫ 1 2 T T − H cos ( n ω 0 t ) d t = 2 T [ H n ω 0 sin ( n ω 0 t ) ] 0 1 2 T + 2 T [ − H n ω 0 sin ( n ω 0 t ) ] 1 2 T T = 2 T [ H n 2 π T sin ( n π ) ] = 0 {\displaystyle {\begin{aligned}a_{n}&={\frac {2}{T}}\int _{0}^{{\frac {1}{2}}T}H\cos(n\omega _{0}t)dt+{\frac {2}{T}}\int _{{\frac {1}{2}}T}^{T}-H\cos(n\omega _{0}t)dt\\&={\frac {2}{T}}\left[{\frac {H}{n\omega _{0}}}\sin(n\omega _{0}t)\right]_{0}^{{\frac {1}{2}}T}+{\frac {2}{T}}\left[{\frac {-H}{n\omega _{0}}}\sin(n\omega _{0}t)\right]_{{\frac {1}{2}}T}^{T}\\&={\frac {2}{T}}\left[{\frac {H}{n{\frac {2\pi }{T}}}}\sin(n\pi )\right]&=0\end{aligned}}}
TODO: finish
![{\displaystyle {\begin{aligned}a_{0}&={\frac {1}{T}}\int _{0}^{{\frac {1}{2}}T}Hdt+{\frac {1}{T}}\int _{{\frac {1}{2}}T}^{T}-Hdt\\&={\frac {1}{T}}\left[Ht\right]{\bigg |}_{t=0}^{{\frac {1}{2}}T}-{\frac {1}{T}}\left[Ht\right]{\bigg |}_{t={{\frac {1}{2}}T}}^{T}\\&={\frac {1}{T}}H{\frac {1}{2}}T-0-\left[{\frac {1}{T}}HT-{\frac {1}{T}}H{\frac {1}{2}}T\right]\\&={\frac {H}{2}}-\left[H-{\frac {1}{2}}H\right]\\&={\frac {H}{2}}-{\frac {H}{2}}\\&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97b47434fad107e427bff0862ed99aca417d74e6)
![{\displaystyle {\begin{aligned}a_{n}&={\frac {2}{T}}\int _{0}^{{\frac {1}{2}}T}H\cos(n\omega _{0}t)dt+{\frac {2}{T}}\int _{{\frac {1}{2}}T}^{T}-H\cos(n\omega _{0}t)dt\\&={\frac {2}{T}}\left[{\frac {H}{n\omega _{0}}}\sin(n\omega _{0}t)\right]_{0}^{{\frac {1}{2}}T}+{\frac {2}{T}}\left[{\frac {-H}{n\omega _{0}}}\sin(n\omega _{0}t)\right]_{{\frac {1}{2}}T}^{T}\\&={\frac {2}{T}}\left[{\frac {H}{n{\frac {2\pi }{T}}}}\sin(n\pi )\right]&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ce7d9bc195d0ef26566c6fa187fb820d766fe9)